3.3 Grid generation
3.3.4 Issues on triangulation
Mesh triangulation and grid refinement play an important role in almost every numerical simulation. Therefore, a lot of different techniques have been developed to achieve suitable computational meshes. The user is basically free to use any tool or method which generates a mesh of accurate quality out of the raw data. This section shall give a short overview on some popular triangulation methods.
BASEMENT does not provide an automated routine for mesh generation. Therefore the plugin BASEmesh for the free and open source geographic information system (GIS) software Quantum GIS (QGIS) was developed (see Section3.3.5).
3.3.4.1 Delaunay triangulation and constrained Delaunay triangulation The Delaunay triangulation is one of the most employed triangulation methods because it optimizes several quality criteria: It maximizes the minimal angle and minimizes the maximum containment circle radius, the maximum enclosing circle radius and the roughness of a piecewise-linear interpolation. It also provides good results regarding the minimization of the maximum angle, but it does not find a global optimum in this case.
A Delaunay triangulation has the following properties. It • is the straight line dual of the Voronoi diagram; • is unique;
• respects the circumcircle criterion;
The circumcircle criterion is respected if the circumcircle of every interior triangle does not contain other points.
This corresponds to say that if ABC + CDA < 180° the empty circle criterion is satisfied. • respects the edge circle property: for each edge exists some point-free circle which
passes through the end points;
• respects the neighbour property: an edge formed by joining a vertex to its nearest neighbour is an edge of the triangulation.
BASEMENT System Manuals 3.3. Grid generation b) B A D C a) B A D C constrained edges
Figure 3.5 a) edge circle criterion. b) empty circle criterion
3.3.4.1.1 Constrained Delaunay triangulation (CDT)
The terrain data is sometimes provided in the shape of a PSLG as it contains break lines which must be conserved as edges in the triangulation. In this case, the constrained Delaunay triangulation can be used.
For the definition of a constrained Delaunay triangulation the notion of visibility of a point is needed: In a PSLG domain P a point D is visible to a point C if the open line segment CD lies within the domain and does not intersect any edges or vertices of P. Point D is visible to CB if it is visible to some point on CB.
For the CDT the edge circle and the empty circle criterion are modified as follows: • Edge circle: for each edge a circle passing through its end-points containing no other
point of the domain visible to the edge exists;
• Empty circle: the circumcircle of every triangle contains no points visible to points inside of the triangle.
3.3.4.1.2 MinMax triangulation
The MinMax triangulation minimizes the maximum angle. It has proven to be very useful in CFD (Barth, 1994).
3.3.4.1.3 Data dependent triangulation
A data dependent triangulation can be used for a 2.5 d problem. Its aim is to find the best triangulation under data dependent constraints, by minimizing a local cost function of a piece-wise interpolation. Two examples of local cost functions are described in (Barth, 1994).
3.3.4.1.4 Steiner triangulation
A Steiner triangulation is a triangulation in which extra points are added to the original data to improve the quality of the mesh. The additional points are called Steiner points. The number of Steiner points must be limited, limiting also the quality of the mesh.
3.3. Grid generation BASEMENT System Manuals
Figure 3.6 a) triangulated half planes. b) merged triangulation
3.3.4.1.5 Non obtuse triangulation
One of the most interesting types of Steiner triangulations is the triangulation without obtuse angles. It is a Delaunay triangulation or constrained Delaunay triangulation and also minimizes the maximum angle and maximizes the minimum height (Bern and Eppstein, 1995).
3.3.4.2 Algorithms
3.3.4.2.1 Sweep-line algorithm
The sweep-line algorithm can be used to perform a first triangulation without any quality criteria. It adds the points by x-coordinate order and then connects them to all visible points of the convex hull of the existing triangulation.
3.3.4.2.2 Divide and conquer
For the divide and conquer algorithm the point set is divided in two half planes along the x axis. Then each half plane is triangulated recursively, and finally the two planes are merged.
3.3.4.2.3 Incremental Insertion algorithms
The incremental insertion algorithms successively insert new points to an existing Delaunay triangulation. These algorithms have a worst case running time of (n2) if the points are
badly ordered. But in practice it is near to 0(nlogn) for Green-Sibson. Watson(or incremental delete and build)
In this algorithm, after the insertion of the point, all the triangles containing the edge q are searched. Then the edges of these triangles visible to q are deleted and the new edge is connected with the vertices of the originated polygon creating new edges.
Green−Sibson
After its insertion, the point q is connected to the vertices of the triangle that contains q. Then all suspect edges have to be swapped to satisfy the empty circle criterion if we want to obtain a Delaunay triangulation. Other edge swapping criteria can be used, for instance the minimization of the maximum angle.
BASEMENT System Manuals 3.3. Grid generation
Figure 3.7 a) point insertion and determination of affected triangles. b) new triangulation
Figure 3.8 a) insertion of the new point b) edge swapping based on empty circle criterion
3.3. Grid generation BASEMENT System Manuals 3.3.4.3 Edge flipping algorithm
The edge flipping algorithm is based on the local optimization of an initial triangulation. For each quadrilateral formed by a convex pair of triangles the diagonal is chosen with regard to a local optimization criterion.
Possible optimization criteria:
• Empty circle criterion: a Delaunay triangulation or constrained Delaunay triangulation is obtained. In this case a global optimum is reached;
• Minimize the maximum internal angle for both triangles: gives a MinMax triangulation only locally optimized;
• Optimize a local cost function: leads to data dependent triangulations (only locally optimal triangulation) (Barth (1994)).
Other properties that can be optimized with edge flipping are for instance vertex degree or total edge length, but a global optimum is not guaranteed. If the algorithm is used to obtain a CDT, the constrained edges simply are not tested because they can not be swapped.
3.3.4.3.1 Edge insertion:
This algorithm solves the problem of minimizing the maximum angle in time O(n2log(n))
exactly. Like the edge flipping algorithm, it starts from an arbitrary initial triangulation. It incidentally inserts candidate edges on a vertex of a worst triangle. The crossed edges are removed, and the remaining regions are re-triangulated. If the triangulation gets worse the added edge is rejected. This algorithm can also be used to find an interpolating surface with minimal slope in time O(n3) (Bern and Plassmann (2000)).
3.3.4.3.2 Blue, red and green refinement
This is another popular method for mesh refinement. Basically, an element with an aspect ratio out of bound gets divided into two elements by inserting a new vertex on the midpoint of the longest edge. This is the so-called green refinement. The new vertex has to be connected on both sides of the edge with the opposing vertices of the neighbouring elements. If one of the two resulting triangles still has a bad aspect ratio, the element gets divided again in the same manner resulting in three new elements. This is called the blue refinement. It is the goal of the blue and green refinement to halve the longest side of the original triangle and whereas degeneration of the mesh in repeated mesh refinement steps is prevented. Finally, the red refinement connects all midpoints of the original edges which leads to four new elements.